Calculus of Variations and Geometric Measure Theory

D. Danielli - R. Ognibene

On a weighted two-phase boundary obstacle problem

created by ognibene on 15 Oct 2021
modified on 27 Mar 2023


Accepted Paper

Inserted: 15 oct 2021
Last Updated: 27 mar 2023

Journal: Indiana University Mathematics Journal
Year: 2021

ArXiv: 2106.13492 PDF


In this work we consider an inhomogeneous two-phase obstacle-type problem driven by the fractional Laplacian. In particular, making use of the Caffarelli-Silvestre extension, Almgren and Monneau type monotonicity formulas and blow-up analysis, we provide a classification of the possible vanishing orders, which implies the strong unique continuation property. Moreover, we prove a stratification result for the nodal set, together with estimates on its Hausdorff dimensions, for both the regular and the singular part. The main tools come from geometric measure theory and amount to Whitney's Extension Theorem and Federer's Reduction Principle.